[{"_1":2,"_3":-5,"_4":-5},"loaderData",{"_5":6,"_7":8},"actionData","errors","root",{"_102":103,"_104":105},"routes/embed.x.$handle.$name.$",{"_9":10,"_11":12,"_13":14,"_15":16,"_17":18,"_19":20,"_21":-5},"handle","jfkll","name","ivmodule","path","numerics.ts","project",{"_22":45,"_11":12,"_33":46,"_47":-5,"_48":42,"_49":50,"_51":50,"_52":53,"_54":-5,"_55":56,"_57":58,"_59":32,"_60":-5,"_61":-5,"_62":63,"_64":65,"_66":-5,"_67":-5,"_68":-5,"_69":-5,"_70":71,"_72":71,"_73":74,"_75":74,"_76":74,"_77":71,"_78":71,"_79":80,"_81":82,"_83":84,"_85":42},"file",{"_22":23,"_11":14,"_24":25,"_26":-5,"_27":28,"_29":30,"_31":32,"_33":34,"_35":36,"_37":38,"_39":40,"_41":42,"_43":-5,"_13":44},"permalink","https://www.val.town/x/jfkll/ivmodule/code/numerics.ts","extension","id","cb090138-a9be-4c9d-8df8-b695c47aa1a4","fileId","570981b0-9e72-11f1-8367-1607ee4eb77e","parentId","branchId","9501ca55-9d64-4f95-91aa-9fa7ca96c333","branchVersion",29,"latest",true,"createdAt",["D",1787434942688],"version",2,"fileType","script","content","/**\n * numerics.ts — zero-dependency numerical core.\n *\n * Everything here is pure and deterministic. No I/O, no Date.now(), no Math.random().\n * That is deliberate: the whole engine must be replayable against historical\n * snapshots for calibration, and Monte Carlo noise on a probability output\n * destroys downstream calibration.\n */\n\n// ---------------------------------------------------------------------------\n// Normal distribution\n// ---------------------------------------------------------------------------\n\nexport function normPdf(x: number): number {\n return Math.exp(-0.5 * x * x) / Math.sqrt(2 * Math.PI);\n}\n\n/** Abramowitz & Stegun 7.1.26-based erf; ~1.5e-7 absolute accuracy. Plenty. */\nexport function normCdf(x: number): number {\n const sign = x < 0 ? -1 : 1;\n const z = Math.abs(x) / Math.SQRT2;\n const t = 1 / (1 + 0.3275911 * z);\n const y = 1 -\n ((((1.061405429 * t - 1.453152027) * t + 1.421413741) * t - 0.284496736) *\n t +\n 0.254829592) *\n t *\n Math.exp(-z * z);\n return 0.5 * (1 + sign * y);\n}\n\n// ---------------------------------------------------------------------------\n// Black-Scholes\n// ---------------------------------------------------------------------------\n\nexport interface BsInputs {\n S: number; // spot / forward\n K: number; // strike\n tau: number; // time to expiry, YEARS\n sigma: number; // implied vol for THIS strike (not ATM)\n r?: number; // carry rate\n}\n\nexport function d1(i: BsInputs): number {\n const r = i.r ?? 0;\n if (i.tau <= 0 || i.sigma <= 0) return i.S > i.K ? 50 : -50;\n return (\n (Math.log(i.S / i.K) + (r + 0.5 * i.sigma * i.sigma) * i.tau) /\n (i.sigma * Math.sqrt(i.tau))\n );\n}\n\n/** Per-contract gamma: d(delta)/dS. */\nexport function bsGamma(i: BsInputs): number {\n if (i.tau <= 0 || i.sigma <= 0) return 0;\n return normPdf(d1(i)) / (i.S * i.sigma * Math.sqrt(i.tau));\n}\n\n/** Charm for a CALL: d(delta)/d(t). Put charm = callCharm (same magnitude, delta differs by constant). */\nexport function bsCharmCall(i: BsInputs): number {\n if (i.tau <= 0 || i.sigma <= 0) return 0;\n const r = i.r ?? 0;\n const D1 = d1(i);\n const D2 = D1 - i.sigma * Math.sqrt(i.tau);\n return (\n -normPdf(D1) * ((2 * r * i.tau - D2 * i.sigma * Math.sqrt(i.tau)) /\n (2 * i.tau * i.sigma * Math.sqrt(i.tau)))\n );\n}\n\n/** Vanna: d(delta)/d(sigma) = d(vega)/dS. */\nexport function bsVanna(i: BsInputs): number {\n if (i.tau <= 0 || i.sigma <= 0) return 0;\n const D1 = d1(i);\n const D2 = D1 - i.sigma * Math.sqrt(i.tau);\n return (-normPdf(D1) * D2) / i.sigma;\n}\n\n// ---------------------------------------------------------------------------\n// Tridiagonal (Thomas algorithm)\n// ---------------------------------------------------------------------------\n\n/**\n * Solves A x = d where A is tridiagonal with sub a, diag b, super c.\n * Arrays are mutated internally on copies; inputs are not modified.\n */\nexport function thomas(\n a: Float64Array,\n b: Float64Array,\n c: Float64Array,\n d: Float64Array,\n): Float64Array {\n const n = b.length;\n const cp = new Float64Array(n);\n const dp = new Float64Array(n);\n\n cp[0] = c[0] / b[0];\n dp[0] = d[0] / b[0];\n for (let i = 1; i < n; i++) {\n const m = b[i] - a[i] * cp[i - 1];\n cp[i] = c[i] / m;\n dp[i] = (d[i] - a[i] * dp[i - 1]) / m;\n }\n\n const x = new Float64Array(n);\n x[n - 1] = dp[n - 1];\n for (let i = n - 2; i >= 0; i--) x[i] = dp[i] - cp[i] * x[i + 1];\n return x;\n}\n\n// ---------------------------------------------------------------------------\n// First-passage PDE\n// ---------------------------------------------------------------------------\n\nexport interface FirstPassageSpec {\n /** Log-price grid, ascending, uniform spacing. x = ln(S/S_ref). */\n x: Float64Array;\n /** Index into x of the current spot. */\n spotIdx: number;\n /** drift(x, tIdx) in log space, per unit of clock time. */\n drift: (x: number, tIdx: number) => number;\n /** volatility(x, tIdx) in log space, per sqrt(unit clock time). */\n vol: (x: number, tIdx: number) => number;\n /** Number of backward time steps. */\n nSteps: number;\n /** Total remaining clock time (same units as drift/vol). */\n tau: number;\n /** Barrier index in x. Absorbing. */\n barrierIdx: number;\n /** true = barrier is above spot (upper), false = below. */\n upper: boolean;\n}\n\n/**\n * Backward Kolmogorov for u(x,t) = P(hit barrier before T | X_t = x).\n *\n * du/dt + mu(x) du/dx + 0.5 sigma^2(x) d2u/dx2 = 0, u(barrier,.) = 1, u(x,T) = 0\n *\n * Crank-Nicolson in time, central differences in space, Thomas solve per step.\n * Returns P(touch before T) evaluated at the spot node.\n */\nexport function firstPassageProb(s: FirstPassageSpec): number {\n const { x, drift, vol, nSteps, tau, barrierIdx, upper, spotIdx } = s;\n\n // Restrict the domain to the side of the barrier the spot lives on.\n const lo = upper ? 0 : barrierIdx;\n const hi = upper ? barrierIdx : x.length - 1;\n const n = hi - lo + 1;\n if (n < 4) {\n return upper\n ? (spotIdx >= barrierIdx ? 1 : 0)\n : (spotIdx <= barrierIdx ? 1 : 0);\n }\n\n const dx = x[1] - x[0];\n const dt = tau / nSteps;\n\n let u = new Float64Array(n); // terminal condition: 0 everywhere except barrier\n const bIdxLocal = upper ? n - 1 : 0;\n u[bIdxLocal] = 1;\n\n const a = new Float64Array(n);\n const b = new Float64Array(n);\n const c = new Float64Array(n);\n const d = new Float64Array(n);\n\n for (let step = nSteps - 1; step >= 0; step--) {\n for (let i = 0; i < n; i++) {\n const gi = i + lo;\n\n // Dirichlet at the absorbing barrier.\n if (i === bIdxLocal) {\n a[i] = 0;\n b[i] = 1;\n c[i] = 0;\n d[i] = 1;\n continue;\n }\n // Far boundary: absorbing at 0 (probability of reaching the barrier from\n // very far away in the remaining time is negligible). Cheap and stable.\n if (i === 0 || i === n - 1) {\n a[i] = 0;\n b[i] = 1;\n c[i] = 0;\n d[i] = 0;\n continue;\n }\n\n const mu = drift(x[gi], step);\n const sg = vol(x[gi], step);\n const v2 = sg * sg;\n\n const A = (0.5 * v2) / (dx * dx); // diffusion coefficient\n const B = mu / (2 * dx); // convection coefficient\n\n // CN: (I - dt/2 L) u^n = (I + dt/2 L) u^{n+1}\n const lSub = A - B;\n const lDiag = -2 * A;\n const lSup = A + B;\n\n a[i] = -0.5 * dt * lSub;\n b[i] = 1 - 0.5 * dt * lDiag;\n c[i] = -0.5 * dt * lSup;\n\n d[i] = u[i] +\n 0.5 * dt * (lSub * u[i - 1] + lDiag * u[i] + lSup * u[i + 1]);\n }\n\n u = thomas(a, b, c, d);\n\n // Probabilities must stay in [0,1]; CN can overshoot marginally near the\n // barrier on the first steps. Clamp rather than switch to fully implicit,\n // which would cost accuracy everywhere to fix a boundary artefact.\n for (let i = 0; i < n; i++) u[i] = u[i] < 0 ? 0 : u[i] > 1 ? 1 : u[i];\n }\n\n const localSpot = spotIdx - lo;\n if (localSpot < 0) return upper ? 0 : 1;\n if (localSpot > n - 1) return upper ? 1 : 0;\n return u[localSpot];\n}\n\n// ---------------------------------------------------------------------------\n// Layer 0 null model — reflection principle\n// ---------------------------------------------------------------------------\n\n/**\n * P(running max of driftless arithmetic BM reaches b>0 before T).\n * This is the benchmark every layer above must beat out-of-sample.\n */\nexport function reflectionTouchProb(\n b: number,\n sigma: number,\n tau: number,\n): number {\n if (tau <= 0 || sigma <= 0) return b <= 0 ? 1 : 0;\n if (b <= 0) return 1;\n return Math.min(1, 2 * normCdf(-b / (sigma * Math.sqrt(tau))));\n}\n\n/** Expected range of driftless BM: 2*sqrt(2/pi)*sigma*sqrt(tau) (Parkinson). */\nexport function expectedRange(sigma: number, tau: number): number {\n return 2 * Math.sqrt(2 / Math.PI) * sigma * Math.sqrt(tau);\n}","deleted",false,"label",[14],"2119d995-ca04-498c-aa95-f7a73f7c0d10",["D",1787415250732],"deletedAt","legacyEnvVars","privacy","public","httpPrivacy","description","","imageUrl","userId","edb0ee8b-6eb5-4033-8065-92fa86e65418","pinnedFileId","0ca60dd2-9e72-11f1-83cf-1607ee4eb77e","pinPreview","pinMigrated","screenshotVersion","likeCount",0,"orgId","019f98c4-b792-72f1-83e7-7d98b5b52c8e","tursoName","tursoHostname","tursoMeasuredAt","pinned","userPerm","r","valPerm","blobPerm","rw","emailPerm","sqlitePerm","projectPerm","telemetryPerm","user",{"_9":10,"_101":53},"org",{"_9":10,"_96":53,"_97":53,"_55":56,"_98":99,"_100":42},"branch",{"_35":86,"_11":87,"_22":28,"_88":45,"_89":-5,"_90":-5,"_91":-5,"_92":-5,"_93":-5,"_94":12,"_95":10},"isOwned",63,"main","projectId","forkedBranchId","forkedBranchVersion","forkedProjectPrivacy","forkedProjectId","forkedProjectOrgId","projectName","userHandle","displayName","avatar","tier","free","canRestrictVals","profileImageUrl","ctx",{"_79":-5,"_135":136,"_137":138,"_139":140,"_141":42,"_142":42,"_143":42,"_144":32,"_145":-5,"_146":-7,"_147":-7,"_148":63,"_149":42,"_150":42,"_151":-5,"_152":42,"_153":42,"_154":42,"_155":-5},"ENV",{"_106":107,"_108":109,"_110":87,"_111":112,"_113":114,"_115":116,"_117":118,"_119":120,"_121":122,"_123":124,"_125":126,"_127":128,"_129":130,"_131":132,"_133":134},"REACT_APP_SENTRY_DSN","https://586020d86f05489db63a554690acf1e9@o1398359.ingest.sentry.io/6724499","RENDER_GIT_COMMIT","e87eb04e707cf64e7e532c8e8ec395f209e17f15","RENDER_GIT_BRANCH","POSTHOG_API_TOKEN","phc_nmoyuAkStEZuUp69rYlg1DZgrIt19Gx0Y4Hwf8tGgJP","POSTHOG_API_HOST","https://wart.val.town","WEB_API_TEMPLATE","https://{val}.web.val.run","APP_URL","https://www.val.town","MODULE_URL","https://esm.town","VANITY_WEB_API_TEMPLATE","https://{label}.val.run/","DENOLS_URL","https://lsp.val.town","NODE_ENV","production","CLOUDFLARE_TURNSTILE_SITE_KEY","0x4AAAAAACtaIZJH7s9rBMZ6","GOOGLE_CLIENT_ID","875989696899-jtsqkhlbdv5jrlsk8631p7ipjg3c6o62.apps.googleusercontent.com","STRIPE_PUBLISHABLE_KEY","pk_live_51N2zvWEpZTTLipIkVEwXrUG9AT0yeOOEYxszarZT1jpsZV58Ly6wlg3SltZpHN26CUalOBmUSY7adQRQiSkCZz2500qSXNkf8Y","API_URL","https://api.val.town","orgs",[],"preferencesKeybindings","default","preferencesTheme","light","preferencesFixers","preferencesCopilot","preferencesSelectionAi","preferencesCloseBrackets","preferencesChatPanelState","preferencesDprint","preferencesDenofmt","notificationCount","flagDenols","isBusinessOrgMember","businessOrgHandle","isBusinessOrgAdmin","isHigherTownieTier","navCollapsed","currentOrgId"]